3.81 \(\int x^5 (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=321 \[ -\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^6 d^3}+\frac{2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^6 d^2}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^6 d}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(8*b*d*x*Sqrt[d - c^2*d*x^2])/(315*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d*x^3*Sqrt[d - c^2*d*x^2])/(945*c^
3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^5*Sqrt[d - c^2*d*x^2])/(525*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (10*b*c
*d*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^6*d) + (2*(d - c^2*d*x^2)^(7/2)*(a
+ b*ArcCosh[c*x]))/(7*c^6*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(9*c^6*d^3)

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Rubi [A]  time = 0.437273, antiderivative size = 366, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5798, 100, 12, 74, 5733, 1153} \[ -\frac{d x^4 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}-\frac{4 d x^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac{8 d (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(8*b*d*x*Sqrt[d - c^2*d*x^2])/(315*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d*x^3*Sqrt[d - c^2*d*x^2])/(945*c^
3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^5*Sqrt[d - c^2*d*x^2])/(525*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (10*b*c
*d*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) - (8*d*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(315*c^6) - (4
*d*x^2*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(63*c^4) - (d*x^4*(1 - c*x)^2*(1 + c*
x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*c^2)

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int x^5 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{8 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac{4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac{d x^4 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (8+20 c^2 x^2+35 c^4 x^4\right )}{315 c^6} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{8 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac{4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac{d x^4 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (8+20 c^2 x^2+35 c^4 x^4\right ) \, dx}{315 c^5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{8 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac{4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac{d x^4 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (8+4 c^2 x^2+3 c^4 x^4-50 c^6 x^6+35 c^8 x^8\right ) \, dx}{315 c^5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{8 d (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac{4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac{d x^4 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}\\ \end{align*}

Mathematica [A]  time = 0.175486, size = 164, normalized size = 0.51 \[ -\frac{d \sqrt{d-c^2 d x^2} \left (35 c^3 x^4 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{4 (c x-1)^{5/2} (c x+1)^{5/2} \left (5 c^2 x^2+2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c}-b \left (\frac{35 c^8 x^9}{9}-\frac{50 c^6 x^7}{7}+\frac{3 c^4 x^5}{5}+\frac{4 c^2 x^3}{3}+8 x\right )\right )}{315 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

-(d*Sqrt[d - c^2*d*x^2]*(-(b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (50*c^6*x^7)/7 + (35*c^8*x^9)/9)) + 35*c^3
*x^4*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + (4*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(2 + 5*c^2*x^
2)*(a + b*ArcCosh[c*x]))/c))/(315*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [B]  time = 0.379, size = 1376, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x)

[Out]

a*(-1/9*x^4*(-c^2*d*x^2+d)^(5/2)/c^2/d+4/9/c^2*(-1/7*x^2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^
(5/2)))+b*(-1/41472*(-d*(c^2*x^2-1))^(1/2)*(256*x^10*c^10-704*c^8*x^8+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+
688*c^6*x^6-576*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-280*c^4*x^4+432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+41*c^2
*x^2-120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(-1+9*arccosh(c*x))*d/(c*x+1
)/c^6/(c*x-1)-1/25088*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+10
4*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-7*(c*x+1)^
(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+7*arccosh(c*x))*d/(c*x+1)/c^6/(c*x-1)+1/3200*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6
-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+13*c^2*x^2-20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)
^(1/2)*(c*x-1)^(1/2)*x*c-1)*(-1+5*arccosh(c*x))*d/(c*x+1)/c^6/(c*x-1)+1/1152*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4
-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))*d/(c
*x+1)/c^6/(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))*d
/(c*x+1)/c^6/(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x)
)*d/(c*x+1)/c^6/(c*x-1)+1/1152*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x
+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))*d/(c*x+1)/c^6/(c*x-1)+1/3200*(-d*(c^2*x^2-1))^(1/2
)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^4-5*(c*x
+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)*(1+5*arccosh(c*x))*d/(c*x+1)/c^6/(c*x-1)-1/25088*(-d*(c^2*x^2-1))^(1
/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-144*c^6*x^6-56
*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(1+7*arccosh(
c*x))*d/(c*x+1)/c^6/(c*x-1)-1/41472*(-d*(c^2*x^2-1))^(1/2)*(-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+256*x^10*
c^10+576*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-704*c^8*x^8-432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+688*c^6*x^6+1
20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-280*c^4*x^4-9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+41*c^2*x^2-1)*(1+9*arccos
h(c*x))*d/(c*x+1)/c^6/(c*x-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.21108, size = 568, normalized size = 1.77 \begin{align*} -\frac{315 \,{\left (35 \, b c^{10} d x^{10} - 85 \, b c^{8} d x^{8} + 53 \, b c^{6} d x^{6} + b c^{4} d x^{4} + 4 \, b c^{2} d x^{2} - 8 \, b d\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (1225 \, b c^{9} d x^{9} - 2250 \, b c^{7} d x^{7} + 189 \, b c^{5} d x^{5} + 420 \, b c^{3} d x^{3} + 2520 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 315 \,{\left (35 \, a c^{10} d x^{10} - 85 \, a c^{8} d x^{8} + 53 \, a c^{6} d x^{6} + a c^{4} d x^{4} + 4 \, a c^{2} d x^{2} - 8 \, a d\right )} \sqrt{-c^{2} d x^{2} + d}}{99225 \,{\left (c^{8} x^{2} - c^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

-1/99225*(315*(35*b*c^10*d*x^10 - 85*b*c^8*d*x^8 + 53*b*c^6*d*x^6 + b*c^4*d*x^4 + 4*b*c^2*d*x^2 - 8*b*d)*sqrt(
-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*c^9*d*x^9 - 2250*b*c^7*d*x^7 + 189*b*c^5*d*x^5 + 420*b*
c^3*d*x^3 + 2520*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 315*(35*a*c^10*d*x^10 - 85*a*c^8*d*x^8 + 53
*a*c^6*d*x^6 + a*c^4*d*x^4 + 4*a*c^2*d*x^2 - 8*a*d)*sqrt(-c^2*d*x^2 + d))/(c^8*x^2 - c^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError